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What does it take for real-deterministic c-valued (i.e., classical,commuting) variables to comply with the Heisenberg uncertainty principle? Here,we construct a class of real-deterministic c-valued variables out of the weakvalues obtained via a non-perturbing weak measurement of quantum operators witha post-selection over a complete set of state vectors basis, which alwayssatisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, weintroduce an auxiliary global random variable and couple it to the imaginarypart of the weak value to transform the incompatibility between the quantumoperator and the basis into the fluctuation of an `error term', and thensuperimpose it onto the real-part of the weak value. We show that this class of``c-valued physical quantities'' provides a real-deterministic contextualhidden variable model for the quantum expectation value of a certain class ofoperators. We then show that the Schr\"odinger and the Kennard-Robertson lowerbounds can be obtained separately by imposing the classical uncertaintyrelation to the c-valued physical quantities associated with a pair ofHermitian operators. Within the representation, the complementarity between twoincompatible quantum observables manifests the absence of a basis wherein theerror terms of the associated two c-valued physical quantities simultaneouslyvanish. Furthermore, quantum uncertainty relation is captured by a specificirreducible epistemic restriction, foreign in classical mechanics, constrainingthe allowed form of the joint distribution of the two c-valued physicalquantities. We then suggest an epistemic interpretation of the two termsdecomposing the c-valued physical quantity as the optimal estimate under theepistemic restriction and the associated estimation error, and discuss theclassical limit.

Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable